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sessionInfo()
R version 4.3.3 (2024-02-29)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.5
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Los_Angeles
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] htmlwidgets_1.6.4 compiler_4.3.3 fastmap_1.1.1 cli_3.6.2
[5] tools_4.3.3 htmltools_0.5.7 rstudioapi_0.15.0 yaml_2.3.8
[9] rmarkdown_2.25 knitr_1.45 xfun_0.42 digest_0.6.34
[13] jsonlite_1.8.8 rlang_1.1.3 evaluate_0.23
library(tidyverse)library(faraway)
1 Parametric vs nonparametric models
Given a regressor/predictor \(x_1,\ldots,x_n\) and response \(y_1, \ldots, y_n\), where \[
y_i = f(x_i) + \epsilon_i,
\] where \(\epsilon_i\) are iid with mean zero and unknown variance \(\sigma^2\).
Parametric approach assumes that \(f(x)\) belongs to a parametric family \(f(x \mid \boldsymbol{\beta})\). Examples are \[\begin{eqnarray*}
f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x \\
f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x + \beta_2 x^2 \\
f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x^{\beta_2}.
\end{eqnarray*}\]
Nonparametric approach assumes \(f\) is from some smooth family of functions.
Nonparametric approaches offers flexible modelling of complex data, and often serve as valuable exploratory data analysis tools to guide formulation of meaningful and effective parametric models.
exa %>%ggplot(mapping =aes(x = x, y = y)) +geom_point() +geom_smooth(span =0.2) +# small span give more wigglenessgeom_line(mapping =aes(x = x, y = m)) # true model (black line)
Moving average estimator \[
\hat f_\lambda(x) = \frac{1}{n\lambda} \sum_{j=1}^n K\left( \frac{x-x_j}{\lambda} \right) Y_j = \frac{1}{n} \sum_{j=1}^n w_j Y_j,
\] where \[
w_j = \lambda^{-1} K\left( \frac{x-x_j}{\lambda} \right),
\] and \(K\) is a kernel such that \(\int K(x) \, dx = 1\). \(\lambda\) is called the bandwidth, window width or smoothing parameter.
When \(x\)s are spaced unevenly, the kernel estimator can give poor results. This is improved by the Nadaraya-Watson estimator \[
f_\lambda(x) = \frac{\sum_{j=1}^n w_j Y_j}{\sum_{j=1}^n w_j}.
\]
Asymptotics of kernel estimators \[
\text{MSE}(x) = \mathbb{E} [f(x) - \hat f_\lambda(x)]^2 = O(n^{-4/5}).
\] Typical parametric estimator has \(\text{MSE}(x) = O(n^{-1})\)if the parametric model is correct.
Choice of kernel. Ideal kernel is smooth, compact, and amenable to rapid computation. The optimal choice under some standard assumptions is the Epanechnikov kernel\[
K(x) = \begin{cases}
\frac 34 (1 - x^2) & |x| < 1 \\
0 & \text{otherwise}
\end{cases}.
\]
Choice of smoothing parameter \(\lambda\). Small \(\lambda\) gives more wiggly curves, while large \(\lambda\) yields smoother curves.
for (bw inc(0.1, 0.5, 2)) {with(faithful, {plot(waiting ~ eruptions, col =gray(0.75))# Nadaraya–Watson kernel estimate with normal kernellines(ksmooth(eruptions, waiting, "normal", bw)) })}
Cross-valiation (CV) choose the \(\lambda\) that minimizes the criterion \[
\text{CV}(\lambda) = \frac 1n \sum_{j=1}^n [y_j - \hat f_{\lambda(j)}(x_j)]^2,
\] where \((j)\) indicates that point \(j\) is left out of the fit.
library(sm)with(faithful,sm.regression(eruptions, waiting, h =h.select(eruptions, waiting)))
Smoothing spline approach chooses \(\hat f\) to minize the modified least squares criterion \[
\frac 1n \sum_i [y_i - f(x_i)]^2 + \lambda \int [f''(x)]^2 \, dx,
\] where \(\lambda > 0\) is the smoothing paramter and \(\int [f''(x)]^2 \, dx\) is a roughness penalty. For large \(\lambda\), the minimizer \(\hat f\) is smoother; for smaller \(\lambda\), the minizer \(\hat f\) is rougher. This is the smoothing spline fit.
The minimizer takes a special form: \(\hat f\) is a cubic spline (piecewise cubic polynomial in each interval \((x_i, x_{i+1})\)).
The tuning parameter \(\lambda\) is chosen by cross-validation (either leave-one-out (LOO) or generalized (GCV)) in R.
with(faithful, {plot(waiting ~ eruptions, col =gray(0.75))lines(smooth.spline(eruptions, waiting), lty =2)})
with(exa, {plot(y ~ x, col =gray(0.75))lines(x, m) # true modellines(smooth.spline(x, y), lty =2)})
with(exb, {plot(y ~ x, col =gray(0.75))lines(x, m) # true modellines(smooth.spline(x, y), lty =2)})
The last example exb shows that automatic choice of tuning parameter is not foolproof.
3.2 Regression splines
The regresison spline fit differs from the smoothing splines in that the number of knots can be much smaller than the sample size.
Piecewise linear splines:
# right hockey stick function (RHS) with a knot at crhs <-function(x, c) ifelse(x > c, x - c, 0)curve(rhs(x, 0.5), 0, 1)
Define some knots for Example A
(knots <-0:9/10)
[1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
and compute a design matrix of splines with knots at these points for each \(x\):
# each column is a RHS function with a specific knot dm <-outer(exa$x, knots, rhs)dim(dm)
[1] 256 10
matplot(exa$x, dm, type ="l", col =1, xlab ="x", ylab="")
Compute and dipslay the regression spline fit.
lmod <-lm(exa$y ~ dm)plot(y ~ x, exa, col =gray(0.75))lines(exa$x, predict(lmod))
We can acheive better fit by using more knots in denser regions of greater curvature.
High-order splines can produce a smoother fit. The bs() function can be used to generate the appropriate spline basis. The default is cubic B-splines.
library(splines)matplot(bs(seq(0, 1, length =1000), df =12), type ="l", ylab="", col =1)
# generate design matrix using B-splines with df = 12lmod <-lm(y ~bs(x, 12), exa)plot(y ~ x, exa, col =gray(0.75))lines(m ~ x, exa) # true modellines(predict(lmod) ~ x, exa, lty =2) # Cubic B-spline fit
4 Local polynomials
Both kernel and spline smoothers are vulnerable to outliers.
Local polynomial method fit a polynomial in a window using robust methods, then the predicted response at the middle of the window is the fitted value. This procedure is repeated by sliding the window over the range of the data.
lowess (locally weighted scatterplot smoothing) or loess (locally estimated scatterplot smoothing) functions in R.
We need to choose the order of polynomial and window width. Default window width is 0.75 of data. Default polynomial order is 2 (quadratic).
Examples.
with(faithful, {plot(waiting ~ eruptions, col =gray(0.75)) f <-loess(waiting ~ eruptions) i <-order(eruptions)lines(f$x[i], f$fitted[i])})
with(exa, {plot(y ~ x, col =gray(0.75))lines(m ~ x) f <-loess(y ~ x) # default span = 0.75lines(f$x, f$fitted, lty =2)# try smaller span (proportion of the range) f <-loess(y ~ x, span =0.22)lines(f$x, f$fitted, lty =5)})
with(exb, {plot(y ~ x, col =gray(0.75))lines(m ~ x) f <-loess(y ~ x) # default span = 0.75lines(f$x, f$fitted, lty =2)# span = 1 means whole span of data (smoothest) f <-loess(y ~ x, span =1)lines(f$x, f$fitted, lty =5)})
Pointwise confidence band is obtained by the local parametric fit for smoothing splines or loess.
ggplot(data = exa, mapping =aes(x = x, y = y)) +geom_point(alpha =0.25) +geom_smooth(method ="loess", span =0.22) +geom_line(mapping =aes(x = x, y = m), linetype =2)
Simultaneous confidence band can be constructed by the mgcv package.
library(mgcv)ggplot(data = exa, mapping =aes(x = x, y = y)) +geom_point(alpha =0.25) +geom_smooth(method ="gam", span =0.22, formula = y ~s(x, k =20)) +geom_line(mapping =aes(x = x, y = m), linetype =2)
5 Wavelets (not covered in this course)
In general, we approximate a curve by a family of basis functions \[
\hat f(x) = \sum_i c_i \phi_i(x),
\] where the basis functions \(\phi_i\) are given and the coefficients \(c_i\) are estimated.
Ideally we would like the basis functions \(\phi_i\) to be (1) compactly supported (adatped to local data points) and (2) orthogonal (fast computing).
Orthogonal polynomials and Fourier basis are not compactly supported.
Cubic B-splines are compactly supported but not orthogonal.
Wavelets are both compactly supported and orthogonal.
Haar basis. The mother wavelet function on [0, 1] \[
w(x) = \begin{cases}
1 & x \le 1/2 \\
-1 & x > 1/2
\end{cases}.
\] Next two members are defined on [0, 1/2) and [1/2, 1) by rescaling the mother wavelet to these two intervals. In general, at level \(j\)\[
h_n(x) = 2^{j/2} w(2^j x - k),
\] where \(n = 2^j + k\) and \(0 \le k \le 2^j\).
library(wavethresh)wds <-wd(exa$y, filter.number =1, family ="DaubExPhase")draw(wds)
